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# Optimal production run time and inspection errors in an imperfect production system with warranty

• This paper considers an imperfect production system to obtain the optimal production run time and inspection policy. Contrary to the existing literature this model considerers that product inspection performs at any arbitrary time of the production cycle and after the inspection, all defective products produced until the end of the production run are fully reworked. Due to some misclassification during inspection, from the inspector's side two types of inspection errors as Type Ⅰ and Type Ⅱ are considered to make the model more realistic rather than existing models. Defective items, found by the inspector, are salvaged at some cost before being shipped. Non-inspected defective items are passed to customers with free minimal repair warranty. The model gives three special cases, where it is found that this model converges over the exiting literature. Some numerical examples along with graphical representations are provided to illustrate the proposed model with comparison with the existing models. Sensitivity analysis of the optimal solution with respect to key parameters of the model has been carried out and the implications are discussed.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation: • • Figure 1.  Plot of expected total cost $C (u_{1}, u_{2}| t^{*}\; = \;2.19)$

Figure 2.  Plot of expected total cost $C (t|u_{1}^{*}\; = \;0.00132, u_{2}^{*}\;=\;0.01023)$

Figure 3.  Impact of holding cost $(C_h)$ on expected total cost $C(t*, u_{1}*, u_{2}*)$

Figure 4.  Impact of salvage cost $(C_s)$ on expected total cost $C (t*, u_{1}*, u_{2}*)$

Figure 5.  Impact of warranty cost $(C_w)$ on expected total cost $C (t*, u_{1}*, u_{2}*)$

Figure 6.  Impact of rework cost $(C_r)$ on expected total cost $C (t*, u_{1}*, u_{2}*)$

Figure 7.  Impact of inspection cost $(C_I)$ on expected total cost $C (t*, u_{1}*, u_{2}*)$

Figure 8.  Impact of restoration cost (r) on expected total cost $C (t*, u_{1}*, u_{2}*)$

Table 1.  Summary of numerical results

 This model $(t^*, u_{1}^*, u_{2}^*)$$C(t^*, u_{1}^*, u_{2}^*)$ Sarkar and Saren $(t^*, u^*)$ $C(t^*, u^*)$ Hu and Zong  $(t^*, u_{1}^*, u_{2}^*)$ $C(t^*, u_{1}^*, u_{2}^*)$ Wang  $(t^*, u^*)$ $C(t^*, u^*)$ (2.04, 0.058, 0.251) ${\$}$8.48 (1.85, 0.064)${\$}$8.90 (2.05, 0.062, 0.239) ${\$}$8.49 (1.83, 0.069)${\$}$8.97

Table 2.  Impact of key parameters on optimal solution

 $C_{h}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ $C_{r}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ 0.1 4.56 0.026 0.112 7.12 2.4 2.14 0.0552 0.15 8.00 0.3 2.64 0.045 0.194 7.92 2.6 2.11 0.0560 0.17 8.16 0.5 2.04 0.058 0.251 8.48 2.8 2.08 0.0568 0.21 8.32 0.7 1.72 0.068 0.296 8.93 3.0 2.04 0.0579 0.25 8.48 0.9 1.52 0.078 0.336 9.32 3.2 1.10 0.0592 0.30 8.62 1.1 1.37 0.086 0.372 9.68 3.4 1.94 0.0608 0.36 8.76 $C_{s}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ 3.6 1.87 0.0631 0.49 8.88 2.0 1.98 0.032 0.382 8.41 $C_{i}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ 2.2 2.01 0.042 0.317 8.44 1.0 2.007 0.003 0.017 8.437 2.4 2.03 0.050 0.278 8.46 1.2 2.041 0.058 0.251 8.479 2.6 2.04 0.058 0.251 8.48 1.4 2.068 0.084 0.214 8.511 2.8 2.05 0.065 0.229 8.49 1.6 2.085 0.105 0.181 8.531 3.0 2.06 0.072 0.211 8.51 1.8 2.093 0.124 0.151 8.542 3.2 2.07 0.079 0.196 8.52 2.0 2.094 0.141 0.120 8.543 $C_{w}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ $r$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ 6.4 2.014 0.123 0.254 8.446 1200 1.94 0.061 0.264 8.35 6.8 2.024 0.105 0.253 8.457 1300 1.97 0.060 0.259 8.40 7.2 2.031 0.089 0.252 8.466 1400 2.01 0.059 0.255 8.44 7.6 2.037 0.073 0.251 8.473 1500 2.04 0.058 0.251 8.48 8.0 2.042 0.058 0.251 8.479 1600 2.07 0.057 0.246 8.52 8.4 2.045 0.041 0.250 8.483 1700 2.11 0.056 0.243 8.56 8.8 2.047 0.018 0.249 8.486 1800 2.14 0.055 0.239 8.60
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Tables(2)

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